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  "body": "\n## Definition\n\nLet $R$ be a [[commutative ring|0GYGHKX834]] and let $A$ be an associative algebra over $R$.\nA *derivation* of $A$ over $R$ is a map $D \\colon A \\to A$ such that:\n\n1. $D$ is $R$-[[linear|0RJF668P1T]]: $D(a + b) = D(a) + D(b)$ and $D(ra) = r D(a)$ for all $a, b \\in A$ and $r \\in R$.\n2. $D(ab) = D(a) b + a D(b)$ for all $a, b \\in A$ (the Leibniz rule).\n\n## Remarks\n\nExample: let $k$ be a field and $A = k[X]$, and let $D$ be the formal derivative, $D(\\sum_i c_i X^i) = \\sum_i i c_i X^{i-1}$.\nIt is $k$-linear, and on monomials $D(X^i X^j) = (i + j) X^{i+j-1} = i X^{i-1} X^j + X^i \\cdot j X^{j-1}$, which gives clause 2 by linearity.\nNon-example: on $k[X]$ with $k$ of characteristic $0$, the second derivative $f \\mapsto f''$ is linear but fails clause 2, since $(X \\cdot X)'' = 2$ while $X'' \\cdot X + X \\cdot X'' = 0$.\nConsequences: $D(1) = 0$, hence $D(r \\cdot 1) = 0$ for all $r \\in R$, and $D(a^n) = n a^{n-1} D(a)$ when $A$ is commutative.\nIf $D_1$ and $D_2$ are derivations, then $D_1 D_2 - D_2 D_1$ is a derivation.\nMore generally, for an $A$-bimodule $M$, a derivation $A \\to M$ is an $R$-linear map with $D(ab) = D(a) b + a D(b)$.\nA derivation of a ring $A$ without a chosen base ring is a derivation over $\\Z$, that is, an additive map with the Leibniz rule.\n",
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